altered to some extent for the adjustment of the finished size distribution. This is an advantage of the “I-system”.
Nomenclature B ( y , x ) =, ( y / ~ =) breakage ~ function, dimensionless CL = circulating load in “E-system”, dimensionless F = make-up feed of system, kg s-l f ( x ) = size frequency distribution of particles within mill, f ~ ( y )= size frequency distribution of feed, m-l f p ( y ) = size frequency distribution of product,
f ( y , t ) = size frequency distribution corresponding to particle size y a t time t , m-l K = constant appearing in selection function, m-n s-l n = exponent appearing in selection function and breakage function, dimensionless Rp(r) = cumulative oversize distribution of product, dimensionless S ( x ) = K x n = selection function, s-l t = milling time, s
W = holdup of mill, kg x = dummy variable for particle size, m x c * = limiting cutoff size in “E-system”, m
Greek Letters y = particle size, m yc = critical size in “I-system”, m Literature Cited Austin, L. G.,Klimpel, R. R., Ind. fng. Chern., 56 (1 l), 18 (1964). Furuya, M., Nakajima, Y . , Tanaka, T., Ind. f n g . Chern., Process Des. Dev., 10,
449 (1971).
National Industrial Research Institute of Kyushu Tosu, Saga-ken, Japan
Norio Ouchiyama’
Department of Chemical Process Engineering Hokkaido Uniuersity Sapporo, J a p a n
Tatsuo Tanaka Yoji Nakajima
Received for reuieu November 17,1975 Accepted April 19, 1976
CORRESPONDENCE
Studies on Tubular Flow Reactor with Motionless Mixing Elements Sir: Jagadeesh and Satyanarayana (1972) have reported an experimental study of pressure drop, residence time distribution (RTD), and reaction between ethyl acetate and sodium hydroxide in an annulus fitted with motionless in-line mixing elements. I t has been concluded that “the presence of such elements results in a reactor which is a nearer approximation to a plug flow reactor even a t low Reynolds numbers”. We would like to raise three points in connection with their paper. First of all, there is an error in their method of calculating dispersion numbers. As a result the influence of mixing elements in narrowing the R T D is much less than that claimed. Secondly, the use of a dispersion model for this system appears unjustified, and thirdly the conversion results are not sufficiently accurate or reproducible to establish that the presence of in-line mixing elements makes the equipment approach a plug flow reactor in performance. Calculation of Dispersion Numbers Jagadeesh et al. calculated dispersion numbers using the expressions
-D= -
DL
u2
2
For discrete data taken a t unequal time intervals, as in their case, the correct expression for variance is (Levenspiel, 1969)
Using eq 3 and 2 dispersion numbers were calculated for the R T D data reported in Table I1 of Jagadeesh et al. The calculated values are compared in Table I with those reported by Jagadeesh et al. I t is clear that the reduction in variance by
introducing in-line mixing elements is about twofold and not five- to ninefold obtained by using the (incorrect) eq 1. Meaningful value of variance requires that the experimental data be accurate, especially toward the two ends of the C curve. Further, the use of variance to calculate dispersion number does not provide a check for the validity of the dispersion model nor does it give a measure of the adequacy of fit. Using the criterion of minimum area between the experimental and the theoretical curves, the best f i t values of dispersion number were calculated numerically. The expression used for the dispersion model solution (Cooper et al., 1971) was
r
1
The calculated best fit values along with the percent error are reported in Table I. This error represents the fraction of the fluid in the exit stream which is assigned incorrect residence time by the model. The reduction in dispersion numbers by using the motionless mixing elements as determined from the best fit values is two- to fivefold. The differences between the dispersion numbers calculated from variance and those numerically obtained with the criterion of minimum error are rather large. The method based on variance would show poorer fit of the experimental data and is, therefore, less satisfactory.
Validity of the Dispersion Model The theoretical curves using the best fit values of the dispersion number are shown along with the experimental data for three of the cases in Figure 1. From the figure the fit appears to he reasonable. However, about 25% of the material Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
473
Table I. Dispersion Numbers Dispersion numbers % error
Best fit values
with best fit values
Flow rate, ml/min
Jagadeesh et al.
Calcd from eq 3 and 2
680 920 1300 1720
0.0085 0.0148 0.0080 0.0070
0.023 0.0214 0.0200 0.0166
0.0213 0.0188 0.0127 0.0106
24.4 23.0 24.8 20.2
500
0.0660
0.0436
0.0520
11.0
Remarks Ann u 1us fitted with mixing elements Empty tube
The limits of the integrand are the radial positions a t which the axial velocities are equal and are obtained from the equation
22 - 2x2 In 2 = ~2 - 2x2 In x
(10)
Integration of eq 9 gives
Using the expressions for U and can be written as
0the dimensionless time 8
Figure 1. Dispersion model fit for annulus. N R ~Theoret. best f i t curve
Exptl data
_._
425
A
__ __
300
200
o
and
Remarks
(13)
Emptytube Annulus fitted with elements
is assigned incorrect residence time by the model. Therefore the suitability of the dispersion model appears questionable.
For an annulus of given cy using eq 8,12,10,and 11,the value of F can be calculated for any 6' > 8min. The expression for C curve is
RTD f o r Diffusion Free L a m i n a r Flow in Annuli For diffusion free laminar flow in an empty annulus the theoretical RTD can be derived by a procedure similar to that used by Danckwerts (1953) for a straight circular tube. This should provide the limiting case against which the effect of introducing mixing elements on R T D may be checked. By definition
F=
J2xUr d r
Q
(5)
For an annulus the point velocity U and the volumetric flow rate Q are given by
where
Using r / R = X, we can write
474
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 3, 1976
-
x (1 - X2 + 2X2 In X) + 22X2 In
x]
(14)
Typical F curves for annuli with different 01 values are shown in Figure 2 along with that for a circular tube. The figure shows that a very thin wire in a large diameter tube, 01 = appreciably decreases the ratio of the maximum t o the mean velocity but has little influence on most of the F curve. With a n increase in cy the R T D is progressively narrowed. Experimental tubular reactors are often provided with axial thermowells and it is interesting to note that such thermowells would bring the reactor closer to plug flow. A comparison of the C curve experimentally obtained by Jagadeesh et al. for the empty tube with that calculated by the above analysis for annulus with cy of 0.64 is shown in Figure 3. The agreement between the two is indeed poor. While it is not clear whether Jagadeesh e t al. obtained empty tube data with an annulus or circular tube the conclusion remains the same since the disagreement with the theoretical F curve for the straight tube will be even more. The variance of the theoretical curve is infinite due to the very long tail. However, if truncated a t 6' of 3.5 and 6.5 the numerically computed variances are 0.19 and 0.45, respectively, as against 0.0436 for the experimental C curve which extends t o 6' of 1.8 only. The nature of the theoretical curve shows that the dispersion model is not suitable t o represent R T D for the empty annulus.
i
I I
4 Figure 2. TheoreticalF curves for empty annulus: - -, = 0.1;-. -, CY = 0.9; --, = 10-9.
CY
= 0; - X -, .~ DIMENSIONLESS TIME
e
Figure 3. C curve for empty annulus: --,
I t is difficult to comment on the exact reason for the discrepancy between the theoretical and the experimental result. The validity of neglecting molecular diffusion in the theoretical analysis can be examined. Thus the limiting cases for the annulus are the circular tube and parallel plates. Of the two, the requirement for negligible influence of molecular diffusion on laminar dispersion is more stringent for the circular tube and is given as T < 0.01 (Gill and Nunge, 1969). For 0.02. the experimental conditions of Jagadeesh et al., T Therefore neglecting molecular diffusion is not unreasonable. On the other hand, for RTD experiments it is necessary that the tracer properties be the same as those of the normal fluid. However, in the case under reference, the density of the tracer was 1.2 g/ml as against 1g/ml for the normal fluid. For laminar dispersion such large density differences can drastically influence dispersion (Gill et al., 1968) and may have contributed to the observed difference. The flow disturbance caused to the laminar flow during tracer injection and possible inaccuracy in manually collecting a number of samples over a short time interval (mean time being 114 s or less) may also have contributed to the discrepancy.
theoretical; 0,experi-
mental data.
-
Annulus Fitted with In-line Mixing Elements The presence of in-line mixing elements is likely to produce a complex flow pattern and may induce some turbulence a t the values of the Reynolds numbers employed. However, in view of the possible shortcomings in the experimental technique pointed out for the empty tube the reliability of the experimental results appears questionable in this case also. This is further supported by experimental work (Nigam et al., 1976) done with a circular tube fitted with in-line mixing elements. The density difference between the tracer and the resident fluid was kept negligible and step input was used t o avoid flow disturbance. The nature of RTD was found t o be drastically different from that represented by the dispersion model and in general similar to that theoretically obtained for the empty annulus. For the results of Jagadeesh e t al. the percent error using the best fit dispersion numbers is high and hence the dispersion model should not be considered suitable for this situation. Annulus with Mixing Elements as a Reactor Jagadeesh et al. conducted a second-order reaction for kfCo up to 0.4 in an annulus fitted with in-line mixing elements. The results were found to lie between the predictions for back
K i to
Figure 4. Deviation of segregated flow annular reactor conversion from plug flow: - -, a = 0; --, CY = 10-9; -, cy = 0.9. - a
mix and plug flow reactors as shown in their Figure 5 . I t was concluded that the presence of in-line mixing elements results in a reactor which is closer to a plug flow reactor in performance. The experimental RTD results were not used for calculating conversion or for checking their experimental results. The appropriate limiting case against which the influence of the in-line mixing elements should be examined is the empty annulus with no molecular diffusion (segregated flow model) and not the back mix reactor. Conversions for the second-order reaction in segregated laminar flow in annuli were calculated using the theoretical F curves. The differences in conversions obtainable in the annulus and plug flow reactor are plotted for two values of CY in Figure 4. The differences in conversions obtainable in circular tube and plug flow reactor are also plotted in Figure 4. For htCo up to 0.5 the predictions for all annuli and straight tubes are nearly the same and the deviation from plug flow reactor conversion is less than -3%. The maximum difference between conversion obtainable in an annulus and plug flow reactor is -5% and occurs for htCo values in the range of 1 to 2. Unless the experiments are inherently very accurate and reproducible, the improvement due to mixing elements is difficult to discern, particularly in the region of hfCo values used by Jagadeech et al. For this region the differences beInd. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
475
C = dimensionless exit concentration for pulse input
D =
axial dispersion coefficient
D , = molecular diffusion coefficient E ( 0 ) = dimensionless point tracer concentration at the outlet E ( @ = dimensionless average tracer concentration at the outlet
F = dimensionless exit concentration for step input
k = specific reaction rate constant L = length of the reactor tx
1 L1
K i c. 0
0
F i g u r e 5. Deviation f r o m p l u g flow reactor conversion: --, segregated laminar flow in e m p t y tube; - -, back m i x reactor; 0,experim e n t a l data.
tween plug flow conversions and those obtainable in back mix and segregated circular tube reactors are plotted in Figure 5. The deviations of experimental conversions from plug flow values are shown as open circles. If a discernible and consistent improvement is affected by the presence of in-line mixing elements the experimental data should lie between the solid curve and the X axis. However, this is not the case. Further, the differences between the results of duplicate experiments are more than the differences between the plug flow conversions and those calculated from the segregated flow in circular tubes, the latter being nearly the same as those for annuli. Conclusions It may be concluded that apart from an error in their method of calculating dispersion numbers, the experimental technique for measuring RTD appears to be inadequate. The applicability of dispersion model for the empty annulus also appears questionable. Any beneficial influence of in-line motionless mixing elements so far as conversion is concerned is also not established. Although the introduction of such elements may be intuitively expected to narrow the RTD this is not the case under the experimental conditions employed. As such there appears to be little justification to build a reactor of this complexity from a practical point of view. Nomenclature
a = radial distance between the point of maximum velocity and the outer radius of the inner tube
U = pressuredrop Q = volumetric flow rate r = radial position in the annulus R = inside radius of outer tube R’ = outside radius of inner tube t_ = time t = mean residence time = axialvelocity U = average velocity X,R = dimensionless radial position for equal axial velocity 2 = conversion
Dimensionless Group N R =~ Reynolds number D / U L = dispersion number Subscripts 0 = initial concentration P = plugflow min = value of 6’ a t which the tracer first appears at the outlet. Greek Letters a = RJR’ = dimensionless time ( t / t )
e
8 = mean dimensionless time = variance X = dimensionless distance a t which the momentum flux is zero I.( = viscosity 7 = characteristic time (iDm/a2) 02
L i t e r a t u r e Cited Cooper, A. R., Jefferys. G. V., “Chemical Kinetics and Reactor Design”, p 283, Oliver and Boyd, 1971. Danckwerts. P. V., Chem. fng. Sci., 2, l(1953). Gill, W. N., Bardhunh, A. J., Reejhsinghani, N. S.. AIChEJ., 14, 100 (1968). Gill, W. N., Nunge, R. J., Ind. Eng. Chem., 61, 33 (1969). Jagadeesh, V., Satyanarayana. M..lnd. Eng. Chem., Process Des. Dev., 11,520
(1972). Levenspiel, O., “Chemical Reaction Engineering”, p 252, Wiley, New Yo&, N.Y.,
1969. Nigam. K. D. P., Vasudeva, K., unpublished work, Department of Chemical Technology, University of Bombay, 1976.
Department of Chemical Technology University of Bombay Bombay 400 019, India
K. D. P. N i g a m K. V a s u d e v a *
Studies on Tubular Flow Reactor with Motionless Mixing Elements Sir: The following are our comments with regard to the criticism raised by Nigam and Vasudeva on our article (Jagadeesh and Satyanarayana, 1972). First, our data are substantially for equal time intervals (Table I1 in the article) and not for unequal time intervals as they claim, and hence the expression for variance 02
=
W E (6’) A0 - ZeE (6’) A0
(1)
ZE(0)AO ZE(0)AO need not be used for calculation of dispersion numbers. The expressions 476
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
u2
= ZPE(6’)~
ZE(0) -=-
D
a2
iJL
2
(2)
(3)
are sufficiently accurate for equal time intervals based on “effective experimental points” chosen to calculate the dispersion numbers reported in our article. There is no error involved in the method of calculation of dispersion numbers as such.
